Compact space

A topological space \((X, \tau)\) is compact if every open cover of \(X\) has a finite subcover: for every family \((U_j)_{j\in J}\) in \(\tau\) with \(X = \bigcup_{j\in J} U_j\), there is a finite subset \(J'\subseteq J\) with \(X = \bigcup_{j\in J'} U_j\).